Yes, though it's only 1 join request. A node doesn't send a packet to a specific packet, it just sends a packet, which is picked up by all gateways in range.
Yup, the node can even store multiple of these sessions (address + security keys) and be joined to multiple networks at the same time.
This is more tricky, as the gateway does not typically have access to the encryption keys (and each network will have / generate its own session keys).